# Jacobian conjecture

Let $F:{\u2102}^{n}\to {\u2102}^{n}$ be a polynomial map, i.e.,

$$F({x}_{1},\mathrm{\dots},{x}_{n})=({f}_{1}({x}_{1},\mathrm{\dots},{x}_{n}),\mathrm{\dots},{f}_{n}({x}_{1},\mathrm{\dots},{x}_{n}))$$ |

for certain polynomials^{} ${f}_{i}\in \u2102[{X}_{1},\mathrm{\dots},{X}_{n}]$.

If $F$ is invertible^{}, then its Jacobi determinant $det(\partial {f}_{i}/\partial {x}_{j})$, which is a polynomial over $\u2102$,
vanishes nowhere and hence must be a non-zero constant.

The *Jacobian conjecture* asserts the converse^{}: every polynomial map
${\u2102}^{n}\to {\u2102}^{n}$ whose Jacobi determinant is a non-zero constant
is invertible.

Title | Jacobian conjecture |
---|---|

Canonical name | JacobianConjecture |

Date of creation | 2013-03-22 13:23:46 |

Last modified on | 2013-03-22 13:23:46 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 5 |

Author | PrimeFan (13766) |

Entry type | Conjecture |

Classification | msc 14R15 |

Synonym | Keller’s problem |